Chapter 7

(a) Find \(\int(x+5)^{2} d x\) in two ways: (i) By multiplying out (ii) By substituting \(w=x+5\) (b) Are the results the same? Explain.

Find the indefinite integrals. $$ \int 3 \sqrt{w} d w $$

Find the exact area enclosed by the curve \(y=x^{2}(1-x)^{2}\) and the \(x\) -axis.

Find \(\int 4 x\left(x^{2}+1\right) d x\) using two methods: (a) Do the multiplication first, and then antidifferentiate. (b) Use the substitution \(w=x^{2}+1\). (c) Explain how the expressions from parts (a) and (b) are different. Are they both correct?

Find the indefinite integrals. $$ \int\left(x^{2}+4 x-5\right) d x $$

A car moves with velocity, \(v\), at time \(t\) in hours by $$v(t)=\frac{60}{50^{t}} \quad \text { miles/hour. }$$ (a) Does the car ever stop? (b) Write an integral representing the total distance traveled for \(t \geq 0\) (c) Do you think the car goes a finite distance for \(t \geq 0\) ? If so, estimate that distance.

Find the indefinite integrals. $$ \int\left(\frac{3}{t}-\frac{2}{t^{2}}\right) d t $$

An island has a carrying capacity of 1 million rabbits. (That is, no more than 1 million rabbits can be supported by the island.) The rabbit population is two at time \(t=1\) day and grows at a rate of \(r(t)\) thousand rabbits/day until the carrying capacity is reached. For each of the following formulas for \(r(t)\), is the carrying capacity ever reached? Explain your answer. (a) \(r(t)=1 / t^{2}\) (b) \(r(t)=t\) (c) \(r(t)=1 / \sqrt{t}\)

(a) Between 1995 and 2005, ACME Widgets sold widgets at a continuous rate of \(R=R_{0} e^{0.15 t}\) widgets per year, where \(t\) is time in years since January 1 . 1995\. Suppose they were selling widgets at a rate of 1000 per year on January 1, 1995. How many widgets did they sell between 1995 and 2005 ? How many did they sell if the rate on January 1,1995 was \(150,000,000\) widgets per year? (b) In the first case (1000 widgets per year on January 1,1995 ), how long did it take for half the widgets in the ten-year period to be sold? In the second case \((150,000,000\) widgets per year on January 1,1995 ), when had half the widgets in the ten-year period been sold? (c) In 2005, ACME advertised that half the widgets it had sold in the previous ten years were still in use. Based on your answer to part (b), how long must a widget last in order to justify this claim?

Find the indefinite integrals. $$ \int e^{2 t} d t $$

Find the indefinite integrals. $$ \int\left(x+\frac{1}{\sqrt{x}}\right) d x $$

Find the indefinite integrals. $$ \int\left(x^{3}+5 x^{2}+6\right) d x $$

Find the indefinite integrals. $$ \int\left(e^{x}+5\right) d x $$

Find the indefinite integrals. $$ \int\left(x^{2}+\frac{1}{x}\right) d x $$

Find the indefinite integrals. $$ \int e^{3 r} d r $$

Find the indefinite integrals. $$ \int \cos \theta d \theta $$

Find the indefinite integrals. $$ \int \sin t d t $$

Find the indefinite integrals. $$ \int 25 e^{-0.04 q} d q $$

Find the indefinite integrals. $$ \int 100 e^{4 x} d x $$

Find the indefinite integrals. $$ \int\left(2 e^{x}-8 \cos x\right) d x $$

Find the indefinite integrals. $$ \int(3 \cos x-7 \sin x) d x $$

Find the indefinite integrals. $$ \int \sin (3 x) d x $$

Find the indefinite integrals. $$ \int x \cos \left(x^{2}+4\right) d x $$

Find the indefinite integrals. $$ \int 6 \cos (3 x) d x $$

Find the indefinite integrals. $$ \int(10+8 \sin (2 x)) d x $$

Find the indefinite integrals. $$ \int(12 \sin (2 x)+15 \cos (5 x)) d x $$

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=f(x)\) and \(F(0)=5\). $$ f(x)=6 x-5 $$

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=f(x)\) and \(F(0)=5\). $$ f(x)=x^{2}+1 $$

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=f(x)\) and \(F(0)=5\). $$ f(x)=8 \sin (2 x) $$

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=f(x)\) and \(F(0)=5\). $$ f(x)=6 e^{3 x} $$

A firm's marginal cost function is \(M C=3 q^{2}+4 q+6\). Find the total cost function if the fixed costs are 200 .